$(\ell_p,\|\cdot\|_{\infty})$ Banach or separable

Question

Is $(\ell_p,\|\cdot\|_{\infty})$ for $1\leq p<\infty$ a Banach or separable space?

is there any fast way of proving it without checking separability or completeness with the usual way?

Answer 1

From the inclusions of sets $$\ell_p\subset c_0\subset \ell_\infty,$$ where $c_0$ denotes the set of convergent sequences and the separability of $(c_0,\lVert \cdot\rVert_\infty)$ is separable we conclude that $(\ell_p,\lVert\cdot\rVert_\infty)$ is separable.

$\ell_p$ endowed with the supremum norm is not a closed subspace of $\ell_\infty$ because its closure for this norm is $c_0\neq \ell_p$.